mercator

THE

MERCATORPROJECTIONS

THE NORMAL AND TRANSVERSE MERCATOR PROJECTIONS ON

THE SPHERE AND THE ELLIPSOID

WITH FULL DERIVATIONS OF ALL FORMULAE

PETER OSBORNE

EDINBURGH

2013

This article describes the mathematics of the normal and transverseMercator projections on the sphere and the ellipsoid with full deriva-tions of all formulae.

The Transverse Mercator projection is the basis of many maps cov-ering individual countries, such as Australia and Great Britain, aswell as the set of UTM projections covering the whole world (otherthan the polar regions). Such maps are invariably covered by a set ofgrid lines. It is important to appreciate the following two facts aboutthe Transverse Mercator projection and the grids covering it:

1. Only one grid line runs true northsouth. Thus in Britain onlythe grid line coincident with the central meridian at 2W istrue: all other meridians deviate from grid lines. The UTMseries is a set of 60 distinct Transverse Mercator projectionseach covering a width of 6in latitude: the grid lines run truenorthsouth only on the central meridians at 3E, 9E, 15E,. . .

2. The scale on the maps derived from Transverse Mercator pro-jections is not uniform: it is a function of position. For ex-ample the Landranger maps of the Ordnance Survey of GreatBritain have a nominal scale of 1:50000: this value is only ex-act on two slightly curved lines almost parallel to the centralmeridian at 2W and distant approximately 180km east andwest of it. The scale on the central meridian is constant but itis slightly less than the nominal value.

The above facts are unknown to the majority of map users. They arethe subject of this article together with the presentation of formulaerelating latitude and longitude to grid coordinates.

Preface

For many years I had been intrigued by the the statement on the (British) OrdnanceSurvey maps pointing out that the grid lines are not exactly aligned with meridians andparallels: four precise figures give the magnitude of the deviation at each corner of themap sheets. My first retirement project has been to find out exactly how these figuresare calculated and this has led to an exploration of all aspects of the Transverse Mercatorprojection on an ellipsoid of revolution (TME). This projection is also used for the UniversalTransverse Mercator series of maps covering the whole of the Earth, except for the polarregions.

The formulae for TME are given in many books and web pages but the full derivationsare only to be found in original publications which are not readily accessible: therefore I de-cided to write a short article explaining the derivation of the formulae. Pedagogical reasonssoon made it apparent that it would be necessary to start with the normal and transverseMercator projection on the sphere (NMS and TMS) before going on to discuss the nor-mal and transverse Mercator projection on the ellipsoid (NME and TME). As a result, thelength of this document has doubled and redoubled but I have resisted the temptation to cutout details which would be straightforward for a professional but daunting for a layman.The mathematics involved is not difficult (depending on your point of view) but it does re-quire the rudiments of complex analysis for the crucial steps. On the other hand the algebragets fairly heavy at times; Redfearn (1948) talks of a a particularly tough spot of workand Hotine (1946) talks of reversing series by brute force and algebraso be warned.Repeating this may be seen as a perverse undertaking on my part. To make this articleas self-contained as possible I have added a number of appendices covering the requiredmathematics.

My sources for the TME formulae are to be found in Empire Survey Review dating fromthe nineteen forties to sixties. The actual papers are fairly terse, as is normal for papers byprofessionals for their peers, and their perusal will certainly not add to the details presentedhere. Books on mathematical cartography are also fairly thin on the ground, moreover theyusually try to cover all types of projections whereas we are concerned only with Mercatorprojections. The few books that I found to be of assistance are listed in the Literature (L) orReferences R but they are supplemented with research papers and web material.

This second edition (2013) adds further material and enters the world of hyperref.

I would like to thank Harry Kogon for reading, commenting on and even checking themathematics outlined in these pages. Any remaining errors (and typographical slips) mustbe attributed to myselfwhen you find them please send an email to the address below.

Peter Osborne

Edinburgh, 2008, 2013

Source files [email protected]

Contents

1 Introduction 9

1.1 Geodesy and the Figure of the Earth . . . . . . . . . . . . . . . . . . . 9

1.2 Topographic surveying . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Cartography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 The criteria for a faithful map projection . . . . . . . . . . . . . . . . . 12

1.5 The representative fraction (RF) and the scale factor . . . . . . . . . . . 13

1.6 Graticules, grids, azimuths and bearings . . . . . . . . . . . . . . . . . 14

1.7 Historical outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.8 Chapter outlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Normal Mercator on the sphere: NMS 21

2.1 Coordinates and distance on the sphere . . . . . . . . . . . . . . . . . . 21

2.2 Normal (equatorial) cylindrical projections . . . . . . . . . . . . . . . . 26

2.3 Four examples of normal cylindrical projections . . . . . . . . . . . . . 29

2.4 The normal Mercator projection . . . . . . . . . . . . . . . . . . . . . 35

2.5 Rhumb lines and loxodromes . . . . . . . . . . . . . . . . . . . . . . . 37

2.6 Distances on rhumbs and great circles . . . . . . . . . . . . . . . . . . 42

2.7 The secant normal Mercator projection . . . . . . . . . . . . . . . . . . 45

2.8 Summary of NMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Transverse Mercator on the sphere: TMS 49

3.1 The derivation of the TMS formulae . . . . . . . . . . . . . . . . . . . 49

3.2 Features of the TMS projection . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Meridian distance, footpoint and footpoint latitude . . . . . . . . . . . 60

3.4 The scale factor for the TMS projection . . . . . . . . . . . . . . . . . 61

3.5 Azimuths and grid bearings in TMS . . . . . . . . . . . . . . . . . . . 61

3.6 The grid convergence of the TMS projection . . . . . . . . . . . . . . . 62

3.7 Conformality of general projections . . . . . . . . . . . . . . . . . . . 64

3.8 Series expansions for the unmodified TMS . . . . . . . . . . . . . . . . 65

3.9 Secant TMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 NMS to TMS by complex variables 71

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Closed formulae for the TMS transformation . . . . . . . . . . . . . . 74

4.3 Transformation to the TMS series . . . . . . . . . . . . . . . . . . . . 75

4.4 The inverse complex series: an alternative method . . . . . . . . . . . . 80

4.5 Scale and convergence in TMS . . . . . . . . . . . . . . . . . . . . . . 82

5 The geometry of the ellipsoid 87

5.1 Coordinates on the ellipsoid . . . . . . . . . . . . . . . . . . . . . . . 87

5.2 The parameters of the ellipsoid . . . . . . . . . . . . . . . . . . . . . . 88

5.3 Parameterisation by geodetic latitude . . . . . . . . . . . . . . . . . . . 88

5.4 Cartesian and geographic coordinates . . . . . . . . . . . . . . . . . . 90

5.5 The reduced or parametric latitude . . . . . . . . . . . . . . . . . . . . 92

5.6 The curvature of the ellipsoid . . . . . . . . . . . . . . . . . . . . . . . 93

5.7 Distances on the ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.8 The meridian distance on the ellipsoid . . . . . . . . . . . . . . . . . . 98

5.9 Inverse meridian distance . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.10 Auxiliary latitudes double projections . . . . . . . . . . . . . . . . . . 102

5.11 The conformal latitude . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.12 The rectifying latitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.13 The authalic latitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.14 Ellipsoid: summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Normal Mercator on the ellipsoid (NME) 111

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2 The direct transformation for NME . . . . . . . . . . . . . . . . . . . . 112

6.3 The inverse transformation for NME . . . . . . . . . . . . . . . . . . . 113

6.4 The scale factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.5 Rhumb lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.6 Modified NME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7 Transverse Mercator on the ellipsoid (TME) 1177.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.2 Derivation of the Redfearn series . . . . . . . . . . . . . . . . . . . . . 121

7.3 Convergence and scale in TME . . . . . . . . . . . . . . . . . . . . . . 126

7.4 Convergence in geographical coordinates . . . . . . . . . . . . . . . . 128

7.5 Convergence in projection coordinates . . . . . . . . . . . . . . . . . . 129

7.6 Scale factor in geographical coordinates . . . . . . . . . . . . . . . . . 129

7.7 Scale factor in projection coordinates . . . . . . . . . . . . . . . . . . . 130

7.8 Redfearns modified (secant) TME series . . . . . . . . . . . . . . . . . 132

8 Applications of TME 1358.1 Coordinates, grids and origins . . . . . . . . . . . . . . . . . . . . . . 135

8.2 The UTM projection . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.3 UTM coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . 137

8.4 The British national grid: NGGB . . . . . . . . . . . . . . . . . . . . 139

8.5 Scale variation in TME projections . . . . . . . . . . . . . . . . . . . . 142

8.6 Convergence in the TME projection . . . . . . . . . . . . . . . . . . . 145

8.7 The accuracy of the TME transformations . . . . . . . . . . . . . . . . 146

8.8 The truncated TME series . . . . . . . . . . . . . . . . . . . . . . . . . 149

8.9 The OSGB series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

8.10 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

A Curvature in 2 and 3 dimensions 153A.1 Planar curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

A.2 Curves in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . 156

A.3 Curvature of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

A.4 Meusniers theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

A.5 Curvature of normal sections . . . . . . . . . . . . . . . . . . . . . . . 159

B Lagrange expansions 163B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

B.2 Direct inversion of power series . . . . . . . . . . . . . . . . . . . . . 164

B.3 Lagranges theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

B.4 Application to a fourth order polynomial . . . . . . . . . . . . . . . . . 165

B.5 Application to a trigonometric series . . . . . . . . . . . . . . . . . . . 166

B.6 Application to an eighth order polynomial . . . . . . . . . . . . . . . . 168

B.7 Application to a modified eighth order series . . . . . . . . . . . . . . . 170

B.8 Application to series for TME . . . . . . . . . . . . . . . . . . . . . . 171

B.9 Proof of the Lagrange expansion . . . . . . . . . . . . . . . . . . . . . 173

C Plane Trigonometry 177

C.1 Trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . 177

C.2 Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

C.3 Gudermannian functions . . . . . . . . . . . . . . . . . . . . . . . . . 181

D Spherical trigonometry 183

D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

D.2 Spherical cosine rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

D.3 Spherical sine rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

D.4 Solution of spherical triangles I . . . . . . . . . . . . . . . . . . . . . . 187

D.5 Polar triangles and the supplemental cosine rules . . . . . . . . . . . . 188

D.6 The cotangent four-part formulae . . . . . . . . . . . . . . . . . . . . . 191

D.7 Half-angle and half-side formulae . . . . . . . . . . . . . . . . . . . . 192

D.8 Right-angled triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

D.9 Quadrantal triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

E Power series expansions 197

E.1 General form of the Taylor and Maclaurin series . . . . . . . . . . . . . 197

E.2 Miscellaneous Taylor series . . . . . . . . . . . . . . . . . . . . . . . . 197

E.3 Miscellaneous Maclaurin series . . . . . . . . . . . . . . . . . . . . . . 198

E.4 Miscellaneous Binomial series . . . . . . . . . . . . . . . . . . . . . . 199

F Calculus of variations 201

G Complex variable theory 205

G.1 Complex numbers and functions . . . . . . . . . . . . . . . . . . . . . 205

G.2 Differentiation of complex functions . . . . . . . . . . . . . . . . . . . 208

G.3 Functions and maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

H Maxima code 215H.1 Common code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

H.2 Lagrange reversion examples . . . . . . . . . . . . . . . . . . . . . . . 216

H.3 Meridian distance and rectifying latitude . . . . . . . . . . . . . . . . . 217

H.4 Conformal latitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

H.5 Authalic latitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

H.6 Redfearn series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

L Literature and links 223

R References and Bibliography 231

X Index 239

9Chapter1Introduction

1.1 Geodesy and the Figure of the Earth

Geodesy is the science concerned with the study of the exact size and shape of the Earth inconjunction with the analysis of the variations of the Earths gravitational field. This com-bination of topics is readily appreciated when one realizes that (a) in traditional surveyingthe instruments were levelled with respect to the gravitational field and (b) in modern satel-lite techniques we must consider the satellite as an object moving freely in the gravitationalfield of the Earth. Geodesy is the scientific basis for both traditional triangulation on theactual surface of the earth and modern surveying using GPS methods.

Whichever method we use, traditional or satellite, it is vital to work with well definedreference surfaces to which measurements of latitude and longitude can be referred. Clearly,the actual topographic surface of the Earth is very unsuitable as a reference surface sinceit has a complicated shape, varying in height by up to twenty kilometres from the deep-est oceans to the highest mountains. A much better reference surface is the gravitationalequipotential surface which coincides with the mean sea level continued under the conti-nents. This surface is called the geoid and its shape is approximately a flattened sphere butwith many slight undulations due to the gravitational irregularities arising from the inhomo-geneity in the Earths crust.

However, for the purpose of high precision geodetic surveys, the undulating geoid is nota good enough reference surface and it is convenient to introduce a mathematically exactreference surface which is a good fit to the shape of the geoid. The surface which has beenused for the last three hundred years is the oblate ellipsoid of revolution formed when anellipse is rotated about its minor axis. We shall abbreviate ellipsoid of revolution to simplyellipsoid in this article, in preference to the term spheroid which is used in much of the olderliterature. (We shall not consider triaxial ellipsoids which do not have an axis of symmetry).The shape and size of the reference ellipsoid which approximates the geoid is usually calledthe figure of the Earth.

Chapter 1. Introduction 10

The earliest accurate determinations of the figure of the earth were made by comparingtwo high precision meridian arc surveys, each of which provided a measure of the distancealong the meridian per unit degree at a latitude in the middle of each arc. Two such mea-surements, preferably at very different latitudes, are sufficient to determine two parameterswhich specify the ellipsoidthe major axis a together with the minor axis b or, more usu-ally, the combination of the major axis with the flattening f (defined below). For example,in the first half of the eighteenth century (from 17341749) , French scientists measureda meridian arc of about one degree of latitude in Scandanavia crossing the Arctic circle(French Geodesic Mission to Finland) and a second arc of about three degrees crossing theequator in Peru ( French Geodesic Mission to Peru) and confirmed for the first time theoblateness of the ellipsoid. (See Clarke (1880), pp 413) More accurate measurements ofa French arc, supervised by Delambre from 1792, determined the meridian quadrant fromequator to pole through Paris, as 5130766 toises (of Peru), the standard of length used inthe measurement of the Peruvian arc. The standard toise bar was held in Paris. The Frenchthen defined the meridian quadrant to be 10,000,000 metres and the first standard metre barwas constructed at a length of 0.5130766 toise. (Clarke (1880), pp 1822).

In 1830 George Everest calculated an ellipsoid using what he took to be the best twoarcs, an earlier Indian Arc surveyed by his predecessor William Lambton and once againthe arc of Peru. As more and longer arcs were measured the results were combined to givemore accurate ellipsoids. For example George Biddell Airy discussed sixteen arcs beforearriving at the result he published in 1830:

a = 6377563.4m b = 6356256.9m f = 1/299.32 [Airy1830] (1.1)

where the flattening f , defined as (a b)/a, gives a measure of the departure from thesphere. Similarly Alexander Ross Clarke used eight arcs to arrive at his 1866 ellipsoid:

a = 6378206.4m b = 6356583.8m f = 1/294.98 [Clarke1866] (1.2)

Modern satellite methods have introduced global ellipsoid fits to the geoid, that for theGeodetic Reference System of 1980 (GRS80) being

a = 6378137m b = 6356752.3m f = 1/298.26 [GRS80/WGS84] (1.3)

There are many ellipsoids in use today and they differ by no more than a kilometre fromeach other, with an equatorial radius of approximately 6378km (3963 miles) and a polarsemi-axis of 6356km (3949 miles) shorter by approximately 22km (14 miles). Note thatmodern satellite ellipsoids, whilst giving good global fits, are actually poorer fits in someregions surveyed on a best-fit ellipsoid derived by traditional (pre-satellite) methods.

1.2 Topographic surveying

The aim of a topographic survey is to provide highly accurate maps of some region ref-erenced to a specific datum. By this we mean a choice of a definite reference ellipsoid


together with a precise statement as to how the ellipsoid is related to the area under survey.For example we could specify how the centre of the selected ellipsoid is related to the cho-sen origin of the survey and also how the orientation of the axes of the ellipsoid are relatedto the vertical and meridian at the origin. It is very important to realize that the choice ofdatum for any such survey work is completely arbitrary as long as it is a reasonable fit tothe geoid in the region of the survey. The chosen datum is usually stated on the final maps.

As an example, the maps produced by the Ordnance Survey of Great Britain (OSGB,1999) are defined with respect to a datum OSGB36 (established for the 1936 re-survey)which is still based on the Airy 1830 ellipsoid which was chosen at the start of the originaltriangulation in the first half of the nineteenth century. This ellipsoid is indeed a good fit tothe geoid under Britain but it is a poor fit everywhere else on the globe so it is not used formapping any other country. The OSGB36 datum defines how the Airy ellipsoid is relatedto the the ground stations of the survey. Originally, in the nineteenth century, the originwas chosen at Greenwich observatory but, for the 1936 re-triangulation no single originwas chosen, rather the survey was adjusted so that the latitude and longitude of 11 controlstations remained as close as possible to their values established in the original nineteenthcentury triangulation.

Until 1983, the United States, Canada and Mexico used the North American datumestablished in 1927, namely NAD27. This is based on the Clarke (1866) ellipsoid tied toan origin at Meades Ranch in Kansas where the latitude, longitude, elevation above theellipsoid and azimuth toward a second station (Waldo) were all fixed. Likewise, muchof south east Asia uses the Indian datum, ID1830, which is based on the Everest (1830)ellipsoid tied to an origin at Kalianpur. The modern satellite ellipsoids used in the WorldGeodetic System suchs WGS72, GRS80, WGS84 are defined with respect to the Earthscentre of mass and a defined orientation of axes. See Global Positioning System.

In all, there are two or three hundred datums in use over the world, each with a chosenreference ellipsoid attached to some origin. The ellipsoids used in the datums do not agreein size or position and a major problem for geodesy (and military planners in particular)is how to tie these datums together so that we have an integrated picture of the worldstopography. In the past datums were tied together where they overlapped but now we canrelate each datum to a single geocentric global datum determined by satellite.

Once the datum for a survey has been chosen we would traditionally have proceededwith a high precision triangulation from which, by using the measured angles and baseline,we can calculate the latitude and longitude of every triangulation station from assumedvalues of latitude and longitude at the origin. Note that it is the latitude and longitude valueson the reference ellipsoid beneath every triangulation station that are calculated and usedas input data for the map projections. It is important to realise that once a datum has beenchosen for a survey in some region of the Earth (such as Britain or North America) thenit should not be altered, otherwise the latitude and longitude of every feature in the surveyregion would have to be changed (by recalculating the triangulation data). But this hasalready happened and it will happen again. For example the North American datum NAD27was replaced by a new datum NAD83 necessitating the recalculation of all coordinates, withresulting changes in position ranging from 10m to 200m. If (when) we use one of the new


global datums fitted by satellite technology as the basis for new maps then the latitude andlongitude values of every feature will change slightly again.

1.3 Cartography

A topographic survey produces a set of geographical locations (latitude and longitude) ref-erenced to some specified datum. The problem of cartography, the representation of thelatitudelongitude data on thedatum by a two-dimensional map. There are an infinite num-ber of projections which address this problem but in this article we consider only the nor-mal (N) and transverse (T) Mercator projections, first on the sphere (S) and then on theellipsoid (E). We shall abbreviate these projections as NMS, TMS, NME and TME: theyare considered in full detail in Chapters 2, 3, 6 and 7 respectively. Formulae (withoutderivations) may also be found in in Map ProjectionsA Working Manual, (Snyder, 1987).

We define a map projection by two functions x( , ) and y( , ) which specify theplane Cartesian coordinates (x,y) corresponding to the latitude and longitude coordinates( , ). For the above projections the fundamental origin is taken as a point O on the equa-tor, the positive x-axis is taken as the eastward direction of the projected equator and thepositive y-axis is taken as the northern direction of the projected meridian through O. Thisconvention agrees with that used in Snyders book but beware other conventions! Manyolder texts, as well as most current continental sources, adopt a convention with the x-axisas north and the y-axis as sometimes east and sometimes west! The convention x-north andy-east is also useful when complex mathematics is used, for example Karney (2011),

1.4 The criteria for a faithful map projection

There are several basic criteria for a faithful map projection but it is important to understandthat it is impossible to satisfy all these criteria at the same time. This is simply a reflectionof the fact that it is impossible to deform a sphere or ellipsoid into a plane without creasesor cuts. (This follows from the Theorema Egregium of Gauss. See Gauss (1827)) Thusall maps are compromises to some extent and they must fail to meet at least one of thefollowing five properties

1. One-to-one correspondence of points. This will normally be the case for large scalemaps of small regions but global maps will usually fail this criterion. Points at whichthe map fails to be one-to-one are called singular points. For example, in the normalMercator projection the poles are singular because they project into lines.

2. Uniformity of point (or local) scale. By point scale we mean the ratio of the distancebetween two nearby points on the map and the corresponding points on the ground.Ideally the point scale factor should have the same value at all points. This criterionis never satisfied. In the Mercator projections the scale is true only on two lines atthe most.


3. Isotropy of point scale. Ideally the scale factor would be isotropic (independentof direction) at any point and as a corollary the shape of any small region would beunalteredsuch a projection is said to be orthomorphic (right shape). By small wemean that, at some level of measurement accuracy, the magnitude of the scale doesnot vary over the small region. This condition is satisfied by the Mercator projections.

4. Conformal representation. Consider any two lines on the surface of the Earth whichintersect at a point P at an angle . Let P and be the corresponding point and angleon the map projection. The map is said to be conformal if = at all non-singularpoints of the map. This has the consequence that the shape of a local feature (suchas a short stretch of coastline or a river) is well represented even though there will bedistortion over large areas. All Mercator projections satisfy this criterion.

5. Equal area. We may wish to demand that equal areas on the Earth have equal areason the projection. This is considered to be politically correct by many proponents ofthe Gall-Peters projection but the downside is that such equal area projections distortshapes in the large. The Mercator projections do not preserve area .

In summary the normal Mercator projection has the properties: (a) there are singularpoints at the poles, (b) the point scale is isotropic (so the map is orthomorphic) but themagnitude of the scale varies with latitude, being true on two parallels at most, (c) theprojection is conformal, (d) the projection does not preserve area. The transverse Mercatorprojection has the properties: (a) there are singular points on the equator, (b) the scaleis isotropic (so the map is orthomorphic) with magnitude varying with both latitude andlongitude, being true on at most two curved lines which cannot be identified with parallelsor meridians, (c) the projection is conformal, (d) the projection does not preserve area.

1.5 The representative fraction (RF) and the scale factor

The OSGB (1999) produces many series of maps of Great Britain. For example there areover two hundred Landranger map sheets which are endorsed with the phrase 1:50,000scale, implying that each 80cm80cm sheet covers an area of 40km40km on the ground.This statement is misleading. To clarify the issue we distinguish two concepts: the repre-sentative fraction and the scale factor.

There are four conceptual steps involved in making a map: (1) a survey produces lati-tude, longitude data on a spherical or ellipsoidal datum; (2) the datum is reduced to a smallmodel, the reduction factor being the representative fraction; (3) the position locations onthe small model are projected (by specified formulae, not simply literally) onto a cylindricalor conical sheet wrapped about the model; (4) the sheet is cut and opened out to give a pla-nar map projection. If, in this construction, the cylinder was tangential to the model at theequator, the map distances on the equator will equal these on the reduced model and we saythat the map scale factor is unity on the equator. The scale factor at other points will vary ina way which is determined by the projection formulae and maintaining a small variation ofthe scale factor over the map is an important criterion in the choice of projection. Note that


the second and third of the conceptual steps may be interchanged. Of course one doesntconstruct physical models, neither does one wrap them in sheets of paper: only one step isneeded from a data-base of locations straight to a printer by way of a computer program.

Returning to the example of the 1:50,000 map series produced by the OSGB we nowinterpret that figure as the constant representative fraction. For the details of the projectionwe have to consult the OSGB (1999) literature where we find that the map scale factor, theratio of nearby distances on the map divided by the corresponding distance on the reduceddatum model, is fixed as 0.9996 on the meridian at 2W and elsewhere varies according toprecise formulae which we shall display later. This implies that 2cm on the map, the spacingbetween the grid lines, represents a true distance on the ground of between 0.9996km and1.0007km. Thus each 80cmx80cm map sheet covers only approximately 40km40km. Theprecise variation of scale factor with position will be calculated in later chapters.

Note the usage that a printed map is large scale when the RF, considered as a mathe-matical fraction, is large and the map covers a small area. The OSGB 1:50000 maps areconsidered to be in this category and the 1:5000 series are of even larger scale. Converselysmall scale maps having a small RF, say 1:1000000 (or simply 1:1M), are used to covergreater regions.

This is an appropriate point to mention the concept of a zero dimension for a map pro-jection. This is the smallest size that can be printed on the map and remain visible to thenaked eye. Before the age of digital maps this was often taken as 0.2mm, corresponding to10m on a 1:50000 map. Thus narrow streams or roads cannot be shown to scale on such amap. Even wide roads, such as motorways, are often shown at exaggerated scales. Moderndigital systems are more powerful since they can show more and more detail as the map iszoomed.

1.6 Graticules, grids, azimuths and bearings

The set of meridians and parallels on the reference ellipsoid is called the graticule. There isno obligation to show the projection of the graticule on the map projection but it is usuallyshown on small scale maps covering large areas, such as world maps. and it is usually omit-ted on large scale maps of small areas. For the OSGB 1:50000 series there is no graticulebut small crosses indicate the intersections of the graticule at 5 intervals on the sheet andlatitude and longitude values are indicated at the edges of the sheet.

The projected map is usually constructed in a plane Cartesian coordinate system butonce again there is no obligation to show a reference grid of lines of constant x and yvalues. In general small scale maps are not embellished with a grid whereas large scalemaps usually do have such a reference grid. The OSGB 1:50000 map sheets have a grid ata 2cm intervals corresponding to a nominal (but not exact) spacing of 1km. Note that anykind of grid may be superimposed on a map to meet a users requirements: it need not bealigned to the Cartesian projection axes, nor need it be a Cartesian grid.

On the graticule the angle between the meridian at any point A and another short lineelement AB is called the azimuth of that line element. Our convention is that azimuths are


measured clockwise from north but other conventions exist. (Occasionally azimuth has beenmeasured clockwise from south!) On a projection endowed with a grid the angle betweenthe grid line through the projected position of A and the projection of the line AB is calledthe grid bearing. This clear distinction in terminology shall be adhered to in this work butit is by no means universal.

On normal Mercator projections the projected graticule is aligned to the underlyingCartesian system so the constant-x grid lines correspond to meridians running north-south.The projection is also constructed to ensure that the azimuth and grid bearing are equal.Therefore a rhumb line, a course of constant azimuth on the sphere, becomes a straightline on the projection.

On the transverse Mercator projections the situation is more complicated. The projec-tion of the graticule is a set of complex curves which, in general, are not aligned to theunderlying Cartesian reference grid: the only exceptions are the equator and the centralmeridian. As a result the constant-x grid lines do not run north-south and the azimuth is notequal to the grid-bearing, instead it is equal to the angle between the projected meridian andthe projected line segment AB. The angle between the projected meridian and the constant-xgrid line is called the grid convergence. On large scale maps of restricted regions it is asmall angle but nonetheless important for high accuracy work. The OSGB 1:50000 mapsheets state the value of the grid convergence at each corner of the sheet.

1.7 Historical outline

Gerardus Mercator (15121594) did not develop the mathematics that we shall presentfor his projection (NMS) in Chapter 2; moreover he had nothing at all to do with threeother projections that now carry his nameTMS, NME, TME. In 1569 he published hismap-chart entitled Nova et aucta orbis terrae descriptio ad usum navigantium ementateaccommadata which may be translated as A new and enlarged description of the Earthwith corrections for use in navigation. His explanation is given on the map-chart:

In this mapping of the world we have [desired] to spread out the surface of the globeinto a plane that the places should everywhere be properly located, not only with re-spect to their true direction and distance from one another, but also in accordance withtheir true longitude and latitude; and further, that the shape of the lands, as they appearon the globe, shall be preserved as far as possible. For this there was needed a newarrangement and placing of the meridians, so that they shall become parallels, for themaps produced hereto by geographers are, on account of the curving and bending ofthe meridians, unsuitable for navigation. Taking all this into consideration, we havesomewhat increased the degrees of latitude toward each pole, in proportion to the in-crease of the parallels beyond the ratio they really have to the equator. (Full text isavailable on Wikipedia at Mercator 1569 world map, Legend 3).

This is an admirably clear statement of his approach. In order that the meridians shouldbe perpendicular to the equator, and parallel to each other, it is first necessary to increasethe length of a parallel on the projection as one moves away from the equator. Since the


circumference of a parallel at latitude is 2piRcos it must be scaled up by a factor of secso that it has the same length as the equator on the projection (2piR). Thus, to guarantee thatan azimuth is equal to its corresponding grid bearing, or equivalently rhumb lines project tostraight lines, it is necessary to increase the meridian scale at latitude by a factor of sec .This leads to a gradual increase in the spacing of the parallels on the projection as againstthe uniform spacing of the equirectangular projection.

Exactly how Mercator produced his map is not known: he left no account of his method.He was familiar with the writing of Pedro Nunes (Randles, 2000) who had showed thatrhumb lines are spirals from pole to pole on the sphere, and he had found means of markingsuch rhumbs on his terrestial globe of 1541. It is possible that he employed mechanicalmeans, using templates, one for each of the principal rhumbs. He could have measured thecoordinates of points on a rhumb and transferred them to a plane chart with the parallelsadjusted so that the rhhumbs became straight lines. On the other hand his latitude scale isfairly accurate and most writers assume that he had some means of calculating the spacings.Ten such methods are discussed by Hollander (2005).

The first to publish an account of the construction of a Mercator chart was a Cambridgeprofessor of mathematics named Edward Wright 1558?1615). His publication entitledThe correction of certain errors in navigation (Wright, 1599) discusses the errors of theequirectangular projection and shows how the angles, at least, are correct in Mercatorschart and goes on to explain the construction of such a chart by using a table of secants. Hepublished a very fine chart based on accurate positions taken from a globe modelled by hiscompatriot Emery Molyneux. For many years thereafter the charts were widely describedas Wright-Molyneux map projection.

In addition to his mathematical derivation of the projection Wright imagined a physicalconstruction:

Suppose a sphericall superficies with meridians, parallels, rumbes, and the whole hy-drographical description drawne thereupon, to be inscribed into a concave cylinder,their axes agreeing in one. Let this sphericall superficies swel like a bladder, (whileit is in blowing) equally always in every part thereof (that is, as much in longitude asin latitude) till it apply, and join itself (round about and all alongst, also towards ei-ther pole) unto the concave superficies of the cylinder: each parallel on this sphericallsuperficies increasing successively from the equinoctial [equator] towards either pole,until it come to be of equal diameter with the cylinder, and consequently the meridiansstill wideening themselves, til they become so far distant every where each from otheras they are at the equinoctial. Thus it may most easily be understood, how a sphericallsuperficies may (by extension) be made cylindrical, . . .

It is easy to see how this works. Mercators projection is constructed to preserve anglesby stretching meridians to compensate exactly for the streching of the parallels. The anglepreserving projection is conformal. Now consider Wrights bladder: it must be infinitely ex-tensible and able to withstand infinite pressure as it slides over the perfectly smooth cylinder.The crucial phrase is swel . . . equally always in every part thereof. Therefore the tensionsover both the initial spherical surface and the final cylindrical surface are uniform, albeitof very different magnitudes. This uniformity guarantees that a crossing of two lines on


the sphere will be at at exactly the same angle on the cylinder. Thus we have generated aconformal projection from the sphere to the cylinder. And there is only one such conformalprojection.

The logarithm function was invented by John Napier in 1614 and numerical tables ofmany logarithmic functions were soon readily available (although analytic Taylor expan-sions of functions had to wait another hundred years). In the 1640s, another English math-ematician called Henry Bond (16001678) stumbled on the numerical agreement betweenWrights tables and those for ln[tan()], as long as was identified with (/2+pi/4). Themathematical proof of the equivalence immediately became noted as an important problembut it was nearly thirty years before it was solved by James Gregory (16381675), IsaacBarrow (16301677) and Edmond Halley (16561742) acting independently. See Halley(1696) These proofs eventually coalesced into direct integration of the secant function aspresented in Chapter 2. The modification of this integration for the ellipsoid (and NME) istrivial.

Having given credit to Wright, Bond and others it is now believed the English math-ematician Thomas Harriot (15601621) was possibly the first to calculate the spacings ofthe Mercator projection. His unpublished works have had to await study until very recently.They contain evidence of a method equivalent to that of Edward Wright and moreover heseems to have devised a formula equivalent to the logarithmic tangent formula derived fromcalculus almost one hundred years later. (Pepper, 1967; Lohne, 1965, 1979; Taylor andSadler, 1953; Stedall, 2000)

The transverse Mercator projection on the sphere was included in a set of seven newprojections published (Lambert, 1772) by the Swiss mathematician and cartographer, Jo-hann Heinrich Lambert. As we shall see in Chapter 3, the derivation of this projection onthe sphere is a straightforward application of spherical trigonometry starting from the nor-mal Mercator result. The generalisation to the ellipsoid was carried out by Carl FriedrichGauss (17771855) in connection with the survey of Hanover commenced in 1818. Hisprojection is conformal and preserves true scale on one meridian: this is the projection weshall term TME.

Gauss left few details of his work and the most accessible account is that of Kruger(1912). For this reason the transverse Mercator projection on the ellipsoid is often calledthe GaussKruger projection. Kruger developed two expressions for the TME projection,one a power series in the longitude difference from a central meridian and a second usinga power series in the parameter which describes the flattening of the ellipsoid. The first ofthese methods was extended to higher order by Lee (1945), and Redfearn (1948) in Britainand by Thomas (1952) in the USA. Their results are used for the OSGB map series andthe UTM series respectively. (There are only sub-millimetre differences). It is this methodwhich is described in this article.

It should be noted that in addition to the representation of the projection by power seriesapproximations an exact solution was devised by E. H. Thompson and published by Lee(1976). That solution and the second solution of Kruger are available in Karney (2011).

Finally, I give the abstract of the 1948 paper by Redfearn. The actual paper is highly


condensed (6 pages!) and although it is available (at a price) it will add nothing to the workpresented here.

The Transverse Mercator Projection, now in use for the new O.S. triangulation andmapping of Great Britain, has been the subject of several recent articles in the EmpireSurpey Review. The formulae of the projection itself have been given by various writ-ers, from Gauss, Schreiber and Jordan to Hristow, Tardi, Lee, Hotine and othernot,it is to be regretted, with complete agreement, in all cases. For the purpose for whichthese formulae have hitherto been employed, in zones of restricted width and in rela-tively low latitudes, the completeness with which they were given was adequate, andthe omission of certain smaller terms, in the fourth and higher powers of the eccentric-ity, was of no practical importance. In the case of the British grid, however, we have tocover a zone which must be considered as having a total width of some ten to twelvedegrees of longitude at least, and extending to latitude 61N. This means, firstly, thatterms which have as their initial co-efficients the fourth and sixth powers of the longi-tude will be of greater magnitude than usual, and secondly that the powers of tan arelikewise greatly increased. Lastly, an inspection of the formulae (as hitherto available)shows a definite tendency for the numerical co-efficients of terms to increase as theterms themselves decrease.

1.8 Chapter outlines

Chapter 2 starts by describing by discussing angles and distances on a sphere of radiusequal to the mean radius of the WGA84 ellipsoid. We consider the class of all normal(equatorial) cylindric projections onto a cylinder tangential to the equator of a sphere andcompare and contrast four important examples. The Mercator projection on the sphere(NMS) is defined as the single member of the class which is such that an azimuth on thesphere and its corresponding grid bearing on the map are equal. This property of conformal-ity is then used to derive the projection formulae by a comparison of infinitesimal elementson the sphere and the plane. Rhumb lines and their properties are defined in detail andcontrasted with great-circles. Secant projections are introduced to control scale variation.

Chapter 3 discusses the transverse Mercator projection on the sphere (TMS). In this casewe are considering a projection onto a cylinder which is tangential to the sphere on a greatcircle formed by a meridian and its continuation, e.g. the meridian at 21E and 159W.These projections are rather unusual when applied to the whole globe but in practice weintend to apply them to a narrow strip on either side of the meridian of tangency which isthen termed the central meridian of the transverse projection. The crux is that by consideringa large number of such projection strips we can cover the whole sphere (except near thepoles) at high accuracy. The derivation of the projection formulae is a straightforwardexercise in spherical trigonometry. An important new feature is that corresponding azimuthsand grid bearings are not equal (even though the transformation remains conformal) andwe define their difference as the grid convergence. Finally we present low order seriesexpansions for the projection formulae.


Chapter 4 is the crunch. Our ultimate aim is to derive the projection equations for thetransverse Mercator projection on the ellipsoid (TME) in the form of series expansions.The only satisfactory way of obtaining these results is by using a small amount of complexvariable theory. This method is complicated by both the geometrical problems of the ellip-soid and also by the fact that we need to carry the series to many terms in order to achieve therequired accuracy. Thus, for purely pedagogical reasons, in this chapter we use the complexvariable methods to derive the low order series solutions for TMS (derived in Chapter 3)from the standard solution for NMS. That it works is encouragement for proceeding withthe major problem of constructing the TME projections from NME.

Chapter 5 derives the properties of the ellipse and ellipsoid. In particular we introduce(a) the principal curvatures in the meridian plane and its principal normal plane, (b) thedistinction between geocentric, geodetic and reduced latitudes, (c) the distance metric onthe ellipsoid and (d) the series expansion which gives the distance along the meridian as afunction of latitude, (e) auxiliary latitudes and double projections.

Chapter 6 derives the normal Mercator projection (NME) on the ellipsoid. The methodis a simple generalization of the methods used in Chapter 2 the only difference being inthe different form of the infinitesimal distance element on the ellipsoid. The results forthe projection equations are obtained in non-trivial closed forms. The inversion of theseformulae is not possible in closed form and we must revert to Taylor series expansions.

Chapter 7 uses the techniques developed in Chapter 4 to derive the transverse Mercatorprojection on the ellipsoid (TME) from that of NME. This derivation requires distinctlyheavy algebraic manipulation to achieve our main result, the Redfearn formulae for TME.

Chapter 8 applies the general results of Chapter 7 to two important cases, namely theUniversal Transverse Mercator (UTM) and the National Grid of Great Britain (NGGB). Theformer is actually a set of 60 TME projections each covering 6 degrees of longitude betweenthe latitudes of 80S and 84N and the latter is a single projection over approximately 10degrees of longitude centred on 2W and covering the latitudes between 50N and 60N. Wethen discuss the variation of scale and grid convergence over the regions of the projectionand also assess the accuracy of the TME formulae by examining the terms of the series oneby one. We find that for practical purposes some terms may be dropped, as indeed they arein both the UTM and NGGB formulae. Finally the projection formulae are rewritten in thecompletely different notation used in the OSGB published formulae (see bibliography).

Appendices There are eight mathematical appendices. Some of these were developed forteaching purposes so they are more general in nature.

A Curvature in two and three dimensions.B Inversion of series by Lagrange expansions.C Plane Trigonometry.D Spherical Trigonometry.E Series expansions.F Calculus of variations.G Complex variable theory.H Maxima code.


Blank page. A contradiction.

21

Chapter2Normal Mercator on the sphere: NMS

Coordinates and distance on the sphere. Infinitesimal elements and the metric.Normal cylindrical projection. Angle transformations and scale factors. Fourexamples of normal cylindrical projections. Derivation of the Mercator pro-jection. Rhumb lines and loxodromes. Distances on the Mercator projection.Secant (modified) normal cylindrical projections.

2.1 Coordinates and distance on the sphere

Basic definitions

The intersection of a plane through the centre of the(spherical) Earth with its surface is a great circle: otherplanes intersect the surface in small circles. The inter-sections of the rotation axis of the Earth with its surfacedefine the poles N and S. The meridians are those linesjoining the poles which are defined by the intersectionof planes through the rotation axis with the surface: themeridians are great circles. The parallels are definedby the intersections of planes normal to the rotation axiswith the surface and the equator is the special case whenthe plane is through the centre: the equator is a greatcircle and other parallels are small circles.

P M

Q

N

p

X

K

Y

Z

Figure 2.1

The position of a point P on the sphere is denoted by an ordered pair ( , ) of latitudeand longitude values. Latitude is the angle between the normal at P and the equatorialplane: it is constrained to the interval of [90, 90] or [pi/2, pi/2] radians. On the sphereany normal to the surface passes through the centre. Longitude is the angle between themeridian through P and an arbitrarily chosen reference meridian (established at Greenwichby the Prime Meridian Conference (1884)): it may be defined on either of the intervals[180, 180] or [0, 360] with radian equivalents [pi, pi] and [0, 2pi] radians respectively.The meridians ( constant), the equator ( = 0) and the small circles ( constant, non-zero)constitute the graticule on the sphere. The figure shows a second point Q with coordinates( + , + ), the meridians through P and Q, arcs of parallels PM, KQ and the greatcircle through the points P and Q.

Chapter 2. Normal Mercator on the sphere: NMS 22

Geographical coordinates are normally given as degrees, minutes and decimal secondsor in degrees and decimal minutes or simply in decimal degrees. In equations, however,all angles must be in radiansexcept where explicitly stated. The unit mil, such that6400mil=2pi radians=360, is sometimes used for small angles, in particular the grid con-vergence defined in Section 3.6. The relations between these units are as follows:

1 rad = 57.29578 = 57 17 44.8 = 3437.75 = 206264.8 = 1018.6mil

1 = 0.0174533 rad, 1 = 0.000291 rad = 0.296 mil, 1 = 0.00000485 rad. (2.1)

1mil = 0.000982 rad = 0.0563 = 3.37 = 202.

Radius of the sphere

The Earth is more accurately represented by an ellipsoid (Chapter 5) with semi-major axis(equatorial radius), a, and semi-minor axis b (often mistakenly called the polar radius) andthere are several choices for the radius, R, of a sphere approximating such an ellipsoid. (Thenotation R will be applied to both the full size approximation or to the sphere reduced bythe representative fraction, RF). The most important possibilities are:

the major axis of the ellipsoid, a,

the tri-axial arithmetic mean, (a+a+b)/3) ,

radius of equal area sphere, (a/

2)[1+ e1(1 e2) tanh1 e]1/2, (Section 5.13),

radius of equal volume sphere, (a2b)1/3.

For other possible choices see Maling 1992 (page 76) and Wikipedia Earth radius For WGS(1984) the values to within 10 cm. are

a = 6,378,137.0m, b = 6,356,752.3m,

triaxial mean radius: 6,371,008.8m,

equal volume radius: 6,371,000.8m,

equal area radius: 6,371,007.2m.

In practical calculations with a spherical model it is acceptable to take the mean radius asR = 6371km (3958 miles). For example this is the value taken by the FAI (InternationalAir Federation). No higher accuracy is required since we must use an ellipsoidal model formore precise calculations. For this radius the circumference is 40,030km (24,868 miles)and the meridional quadrant (pole to equator) is 10,007km (6217 miles). One degree oflatitude corresponds to 111.2km. and one minute of latitude corresponds to 1853m.


Cartesian coordinates

If the radius of a parallel circle is p() = Rcos the Cartesian coordinates of P are

X = p()cos = Rcos cos ,Y = p()sin = Rcos sin ,Z = Rsin , (2.2)

with inverse relations

= arctan(

Zp

)= arctan

(Z

X2+Y 2

), = arctan

(YX

). (2.3)

For a point at a height h above the surface at P we simply replace R by R+ h in the directtransformations: the inverse relations for and are unchanged but they are supplementedwith the equation

h =

X2+Y 2+Z2R. (2.4)The unit vector, n, from the centre of the sphere toward a point on the surface is

n = (cos cos , cos sin , sin). (2.5)

Distances on the sphere

In Figure 2.1 the distance PQ in three dimensions is unique but the distance on the surface ofthe sphere depends on the path taken between the points. For two points in general positionthe important distances are along a great circle, which is the shortest distance, or alonga rhumb line which, by definition, intersects meridians at constant azimuth. (Rhumb linedistances are discussed in Section 2.5). For example, if the points are at the same latitude wecan calculate the rhumb distance between them by measuring along the parallel circle; theshorter great circle distance deviates north (south) in the northern (southern) hemisphere.

The only trivially calculated distances are those measured along meridians or parallels:on the meridian in Figure 2.1 we have PK = R and on the parallel PM = p() =Rcos (where and are in radians). For widely separated points these becomePK = R(21) and PM = Rcos (21). It is useful to have some feel for the distanceson meridians and parallels on the sphere (radius 6371km). To the nearest metre:

radius (m) circumference 1 1 1

meridian 6,371,008 40,030,173 111,195 1853 31

equator 6,371,008 40,030,173 111,195 1853 31

parallel at 15 6,153,921 38,666,178 107,406 1790 30parallel at 30 5,517,454 34,667,147 96,297 1604 27parallel at 45 4,504,983 28,305,607 78,626 1310 22parallel at 60 3,185,504 20,015,096 55,597 927 15parallel at 75 1,648,938 10,360,571 28,779 480 8

Table 2.1


One minute of arc on the meridian (of a spherical Earth) was the original definition ofthe nautical mile (nml). On the ellipsoid this definition of the nautical mile would dependon latitude and the choice of ellipsoid therefore, to avoid discrepancies, the nautical mile isnow defined by international treaty as exactly 1852m (1.151 miles). The original definitionremains a good rule of thumb for approximate calculations but note that it corresponds to aspherical Earth model of radius equal to 6366.7km rather than the value of 6371km whichwe used for the previous table.

The great circle distance, g12 between two points in general position is R times theangle (in radians) which the circular arc between them subtends at the centre. That angle isdefined by the two unit vectors giving the positions of the end points.

n1 = (cos1 cos1, cos1 sin1, sin1),n2 = (cos2 cos2, cos2 sin2, sin2),

n1n2 = cos1 cos2 cos(21)+ sin1 sin2g12 = R cos1 [cos1 cos2 cos(21)+ sin1 sin2] . (2.6)

This formula is not well conditioned at short distances and alternative forms are preferable.(See Wikipedia Great-circle distance). There are a number of great circle distance calcula-tors available on the web. The FAI use the mean sphere (and WGS84). The Ed WilliamsAviation page has a more comprehensive list of Earth models. The csgnetwork uses a modela model with radius 6366.7km: it may also be used to calculate way point values.

Infinitesimal elements

In practical terms an element of area on the sphere can be said to be infinitesimal if, fora given measurement accuracy, we cannot distinguish deviations from the plane. To bespecific, consider the spherical element PMQK shown in Figure 2.1, and in enlarged formin Figure 2.2a, where the solid lines PK, MQ, PQ are arcs of great circles, the solid

Figure 2.2

lines PM and KQ are arcs of parallel circles and the dashed lines are straight lines inthree dimensions. From Figure 2.2b, for (rad) 1 the arcchord difference is

arc(AB)AB = R2Rsin 2= R2R

(2 1

3! 3

8+

)=

R 3

24+O(R 5). (2.7)

Suppose the accuracy of measurement is 1m. Setting = we see that the differencebetween the arc and chord PK will be less than 1m, and hence undetectable by measurement,


if we take < (24/R)1/3 0.0155rad, corresponding to 53 or a meridian arc length of99km. Similarly, setting = and replacing R by p= Rcos , the difference between thearc and chord PM at a latitude of 45 (where cos = 1/

2) is less than 1m if is less

than 59, corresponding to an arc length of 78km on that parallel. If we take our limitingaccuracy to be 1mm the above values become 9.9km and 7.8km. A surface element ofthis order or smaller can therefore be well approximated by a planar element which can bemapped without need for projection.

Figure 2.3

We now prove that the small surface element PKQM may be well approximated bya rectangular element. Figure 2.3a shows the planar trapezium which approximates thesurface element. Since PM = Rcos we have

KQPM = dd

(Rcos ) =Rsin . (2.8)

Now the distance PK R so the small angle , called spherical convergence (note [1] )is given by

sin = PMKQ2

1PK

=12

sin . (2.9)

Clearly becomes arbitrarily small as Q approaches P and the infinitesimal element isarbitrarily close to the rectangle with sides R and Rcos shown in Figure 2.3b. Theplanar geometry of the right angled triangle PQM in that figure gives two important resultsfor the azimuth and distance PQ:

tan = limQP

Rcos R

= cosdd

, (2.10)

s2 = PQ2 = R2 2+R2 cos2 2. (2.11)

The latter result also follows directly from equations (2.2):

X =(Rsin cos ) (Rcos sin )Y =(Rsin sin ) +(Rcos cos )Z = (Rcos) , (2.12)

s2 = X2+Y 2+Z2,

s2 = R2 cos2 2+R2 2. (2.13)

This expression defines the metric on the surface of the sphere.


2.2 Normal (equatorial) cylindrical projections

The normal, or equatorial, aspect of cylindrical projections of a (reduced) sphere of ra-dius R are defined on a cylinder of radius R which is tangential to the sphere on the equatoras shown in Figure (2.4). (When the cylinder is tangential to a meridian the aspect is saidto be transverse: for other orientations the aspect is oblique.) The axis of the cylindercoincides with the polar diameter NS and the planes through this axis intersect the spherein its meridians and intersect the cylinder in its generators.

Figure 2.4: The normal cylindrical projection

The projection takes points on the meridian to points on the corresponding generator ofthe cylinder according to some formula which is NOT usually a geometric constructioninparticular the Mercator projection is not generated by a literal projection from the centre (asstated on many web sites): see the next section. The cylinder is then cut along a generatorwhich has been taken at = 180 in Figure 2.4 but could have been chosen at any longitude.Finally the cylinder is unrolled to form the flat map. Note that the last step of unrolling intro-duces no further distortions. Axes on the map are chosen with the x-axis along the equatorand the y-axis coincident with one particular generator, taken as the Greenwich meridian( = 0) in Figure (2.4). Clearly the meridians on the sphere map into lines of constant xon the projection so the x-equation of the projection is simply x = R (radians). For they-equation of the projection we admit any sensible function of , irrespective of whetheror not there is a geometrical interpretation. Therefore normal cylindrical projections aredefined by (note [2] )

x( ,) = R , (2.14)y( ,) = R f (), (2.15)

where and are in radians. With transformations of this form we see that the parallels onthe sphere ( constant) project into lines of constant y so that the orthogonal intersectionsof meridians and parallels of the graticule on the sphere are transformed into orthogonalintersections on the map projection. The spacing of the meridians on the projection isuniform but the spacing of the parallels depends on the choice of the function f ().

Note that all normal cylindrical projections have singular points: the points N, S at


the poles transform into lines given by y = R f (pi/2). On the sphere meridians intersectat the poles but on normal cylindrical projections meridians do not intersect. All otherpoints of the sphere are non-singular points. Of course there is nothing special about thepoles; if we use oblique or transverse aspects the geographic poles are regular points andother points become singularthe singularities at the poles are artifacts of the coordinatetransformations. For example we shall find that the transverse Mercator projection hassingular points on the equator.

The equations (2.14, 2.15) define a projection to a map of constant width, W , equal tothe length of the equator, 2piR. Since the true length of a parallel is 2piRcos , the scalefactor, the map projection length divided by the corresponding true length (on the reducedsphere), along a parallel is equal to sec : this factor increases from 1 on the equator toinfinity at the poles. Note that this statement about scale on a parallel applies to any normalcylindrical projection but the scale on the meridians, and other lines, will depend on f ().

The actual printed projection in Figure 2.4 has a value of W approximately equal to8cm, corresponding to R = 1.27cm and an RF (representative fraction) of approximately 1over 500 million or 1:500M.

Angle transformations on normal cylindrical projections: conformality

In Figure 2.5 we compare the rectangular infinitesimal element PMQK on the sphere withthe corresponding rectangular element PMQK on the projection. We define the angle on the sphere to be the azimuth and the corresponding angle on the projection to be thegrid bearing. This distinction in terminology is not widespread.

Figure 2.5

The geometry of the rectangular elements gives

(a) tan =Rcos

Rand (b) tan =

xy

=

f (), (2.16)

so thattan =

secf ()

tan. (2.17)

If = it follows that the angle between any two azimuths is equal to the angle betweenthe corresponding grid bearings. In this case we say that the projection is conformal. Thecondition for this is that f () = sec : this is the basis for the derivation of the Mercatorprojection given in Section 2.4.


The point scale factor

Define , the point scale at the point P on the projection by

= limQP

distance PQ on projectiondistance PQ on sphere

(2.18)

= limQP

x2+y2

R2 cos2 2+R2 2. (2.19)

Point scale factors on meridians (h) and parallels (k)

When PQ lies along the meridian and x are zeroand y= R f (). The scale factor in this case is conven-tionally denoted by h. Therefore (2.19) gives

meridian scale: h() = f (). (2.20)

On a parallel and y are zero and x = R . Thescale factor in this case is conventionally denoted by k.

parallel scale: k() = sec . (2.21)

The parallel scale factor, plotted alongside, is the samefor all normal cylindrical projections.

k()

0 30 60 900

3

6

9

12

Figure 2.6Point scale factor in a general direction: isotropy of scale

Equations (2.16) give =cot cos and y=cot x. Therefore equation (2.19)gives the scale factor at azimuth as

() = limQP

x2(1+ cot2 )

R2 cos2 2(1+ cot2), (2.22)

= sec[

sinsin (,)

], (2.23)

where (,) can be found from Equation (2.17). For a conformal projection with = the general scale factor is equal to sec , independent of : it is isotropic.Area scale factor The area scale is obtained by comparing the areas of the two rectanglesPMQK and PMQK. Denoting this scale factor by A and using (2.20) and (2.21).

A() = limQP

xy(Rcos )(R)

= sec f () = hk. (2.24)

NB. All of these scale factors apply only to the normal cylindrical projections. They areindependent of , a reflection of the rotational symmetry.


2.3 Four examples of normal cylindrical projections

Consider the following projections of the unit sphere, R = 1:

1. The equirectangular (or equidistant or Plate Carree) projection: f () = .2. Lamberts equal area projection: f () = sin .3. Mercators projection: f () = ln [tan(/2+pi/4)]. (Derived in Section 2.4).4. The central cylindrical projection: f () = tan .

The following table summarizes the properties of these projections.

equirectangular equal-area Mercator centralx x-range (pi, pi) (pi, pi) (pi, pi) (pi, pi)y = f ( ) sin ln[tan(/2+pi/4)] tany-range (pi/2, pi/2) (1, 1) (,) (,)f ( ) 1 cos sec sec2 meridian h 1 cos sec sec2 parallel k sec sec sec secequator k 1 1 1 1area (hk) sec 1 sec2 sec3 (2.17): tan = sec tan sec2 tan tan cos tanaspect ratio 2 pi 0 0

Figure 2.7 Figure 2.8 Figure 2.10 Figure 2.11

Table 2.2

The four projections are shown in Figures (2.72.11). The maps all have the samex-range of (pi, pi) (on the unit sphere) but varying y-ranges. They are portrayed on theprinted page with a width of 12cm corresponding to an RF of approximately 1/300M. Eachof the projections is annotated on the right with a chequered column corresponding to 55regions on the sphere. The width of these rectangles is the same for all projections but theirheight depends on f ().

On the equator = 0 so that both cos and sec are equal to unity. All the scalefactors are unity and equation (2.17) shows that = where any line segment crosses theequator. The projections are perfectly well behaved and quite suitable for accurate largescale mapping close to the equator: both relative distance and shape are well preserved ina comparison with their actual representations on the sphere. Look at an actual globe andcompare all four projections in locations such as Africa, the Caribbean and Indonesia.

Away from the equator all the projections have a parallel scale factor equal to sec :a necessary consequence of the attempt to project the spherical surface onto a rectangulardomain. This factor increases to infinity as pi/2 so that the poles of the sphereare stretched out to lines across the full width of the projection, at finite or infinite values


of the y-coordinate. The poles are singular points of the projection where the one-to-onecorrespondence between sphere and projection breaks down. The horizontal stretching athigh latitudes is leads to distortions in all four projections when they are compared with anactual globe. The shape of Alaska is good measure of this distortion. Only the Mercatorprojection preserves good local shape.

Relative area is another good criterion in assessing the projections. On the globe thearea of Greenland is 1/8 that of South America and 1/13 that of Africa. Only the Lambertequal-area projection preserves these values.

pi/2

1

y=0

1

pi/2

pi pi/2 x=0 pi/2 pi-90

-60

-30

=0

30

60

90-180 -90 =0 90 180

Figure 2.7: Equirectangular projection (R=1)

The equirectangular projection: f () =

This projection, the simplest of all, has been in use since the time of Ptolemy (83?161AD)who attributes its first use to Marinus (Snyder, 1993, page 6). The meridians and parallelsof latitude are equidistant parallel lines intersecting at right angles. (It becomes a rectan-gular grid when applied to a secant projection; see Section 2.7). The overall aspect ratio(width:height) is 2. The projection is also known as the Plate Carree, the Plane Chartor the equidistant projection. The projection was very important in the sixteenth centurybecause it was used to underpin the new portolans and charts which were being extendedto cover ocean sailing as against the local European sailing of earlier times. The projectionis still useful for many applications where the metric properties are irrelevant, for exampleindex map sheets for all sorts of topics.

Like all normal cylindrical projections the equirectangular is well behaved near theequator but as we move away from the equator the failings of its simplicity become morepronounced. Note first Equation 2.17 reduces to tan = cos tan so that 6= unless = = 0 or pi/2. The projection is certainly not conformal and the attractive compassroses shown on early charts give incorrect values for the corresponding azimuths on the


sphere. This may be of no concern for short journeys in equatorial regions but it is verysignificant on long oceanic voyages and at extreme latitudes, even if the sailors of the six-teenth century were aware of this deficiency and had evolved rules of thumb to compen-sate. Note that the errors are not small: if we take = 45 and a latitude = 30 then = arctan

3/2 = 40.9 but at latitude = 60 and = arctan0.5 = 26.6

The alternative name equidistant is also misleading because the scale factor is uni-form, equal to 1, only on the equator and the meridians where the true distance is equal tothe ruler distance divided by the RF. On a parallel we must first divide by a factor of sec .Along any other line in the projection the scale factor depends on both the grid bearingand the latitude: from Equation (2.23) we have = sec sin( ,)cosec where, fromEquation (2.17), = arctan [tan cos ]. It is then possible to relate elements of length, dson the projection and ds on the sphere, and even construct an integral for finite segmentsbut there is basically no point in doing so since the path on the sphere corresponding tothis straight line on the projection is neither a rhumb nor a great circle. It cant be a rhumbbecause the azimuth, is not constant. It is, moreover, a curve on the sphere which, if weassume that it starts at = = 0, attains the pole. For example if the line on the projectionis = 3 it reaches the pole on the sphere when = pi/6. But the only great circles throughthe pole are the meridians so the path on the sphere cannot be a great circle. For the truegreat circle distance between general points on the equirectangular projection we must usethe standard geodesic formulae of equation 2.6.

1

y=0

1

pi pi/2 x=0 pi/2 pi-90-60

-30

=0

30

6090

-180 -90 =0 90 180

Figure 2.8: Lambert equal area projection (R=1)

Lamberts equal area projection: f () = sin

This projection (Lambert, 1772) is constructed to guarantee the equality of correspondingarea elements on the (reduced) sphere and the projection. Since the parallel scale factor isstill as k = sec we must have a scale factor on the meridian given by h = cos . It followsfrom Equation eq:02nms:20 that f () = sin . Thus the ratio of the area of Greenland tothat of Africa is correctly portrayed (as 1/13). In common with all cylindrical projections itis well behaved with good shapes and distances near the equator but it is distorted at highlatitudes: look at look at Alaska for example. The projection is not conformal.

There are problems with angles, exactly as for the equirectangular projection. We now


have tan = cos2 tan so that if we again take = 45 and a latitude = 30 then = arctan0.75 = 36.9 and at latitude = 60 we have = arctan0.25 = 14.0. The gridbearings are completely unreliable.

Once again, interpreting ruler distances on the projection is trivial only on the equatorand parallels (where we must first divide by a factor of sec ). This time a ruler distanceon the meridian has no simple interpretation because of the of the varying scale factorh = cos . However if we measure the ruler distances of two points on a meridian fromthe equator, say y1 and y2, not just their separation, we can then use y = sin to find thecorresponding latitudes, 1 and 2: the length y2 y1 then corresponds to a distance on thesphere equal to R(2 1). On oblique lines of the projection we have exactly the samedifficulties that we encountered with the equirectangular projection.

The projection is one of the few which admits of a geometricinterpretation because y = Rsin is simply the distance NPof a point P at latitude above the equatorial plane. P is pro-jected from the sphere to cylinder along the line KPP parallelto the equatorial plane drawn from the axis of the spherenotprojected from the origin. Thus any narrow (in longitude) stripof the map is basically the view of a classroom globe from adistant side position. Figure 2.9

Mercators projection: f () = ln[tan(/2+pi/4)]

Figure 2.10 shows Mercators projection. Since f (pi/2) = the projection extends to in-finity in the y direction and the map must be truncated at an arbitrarily chosen latitude.Here we have chosen = 85.051 so that the aspect ratio is equal to 1. Truncation atthese high latitudes emphasizes the great distortion near the poleswitness the divergingarea of Antarctica and Greenland as big as Africa. Note that the original (Mercator, 1569)projection was truncated asymmetrically and as a result Europe, already at a larger scalethan Africa, moved nearer to the centre: a source of controversy in the twentieth century(Monmonier, 2004).

The fundamental property of the Mercator projection is that it is conformal, i.e. it is anangle preserving projection. This follows from equation (2.17) since f () = sec impliesthat = . One important corollary is that a rhumb line on the sphere, which crosses theconverging meridians on the sphere at a constant angle , projects into a straight line ofconstant grid bearing on the projection where the meridians are parallel verticals. This isdiscussed more fully in Section 2.5.

Conformality implies isotropy of scale: meridian scale (h), parallel scale (k) and gen-eral scale ( ) are all equal to sec in the Mercator projection. Therefore a small regionof the sphere is projected with very little change of shape, hence the use of the term ortho-morphic projection, (greek: right shape). Witness the realistic shape of small islands farfrom the equator, for example Iceland or Great Britain: larger regions such as Greenland orAntarctica are distorted because the scale factor changes markedly over their extent.


pi3

2

1

y=0

1

2

3pi


-80

-70

-60

-30

=0

30

60

70

80

85-180 -90 =0 90 180

Figure 2.10: Mercator projection truncated at 83.05(R=1)

The words conformality and orthomorphism are often used interchangeably: this ismisleading. Conformality is an exact local (point) property: the angle between two linesintersecting at a point on the sphere is the same on the projection. Orthomorphism is anapproximate non-local property because the shape of small elements is preserved only tothe extent that the latitude variation of scale is undetectable: this depends on the accuracyof measurement. Conformality and low scale distortion near the equator, where sec isapproximately unity, means that the Mercator projection is suitable for accurate large scalemapping near the equator. This is discussed more quantitatively in Section 2.7. It is quiteinappropriate for small scale projections of the world, or large regions of the world, oceaniccharts excepted.

Ruler distances on the equator and parallels can be simply related to true distances, as inthe previous projections, but there are no simple interpretations for ruler distances of otherlines on the projection. Since straight lines on the projection correspond to rhumb linesthese distances are important and we shall return to their calculation in Section 2.6.


Central cylindric projection: f () = tan

pi3

2

1

y=0

1

2

3pi


-70

-60

-30

=0

30

60

70

72-180 -90 =0 90 180

Figure 2.11: Central cylindrical projection truncated at 72(R=1)

The only reason for including the central cylindric projec-tion, a direct projection from the centre of the sphere withy = NP = R tan , is that it is often often claimed to show theconstruction of the Mercator projection. This is of course com-pletely wrong: the Mercator projection is NOT constructed inthis way. The projection is shown truncated at 72.34 to giveunit aspect ratio. The central projection is completely lackingin any virtues and it has never been used for any practical map-ping. We shall not consider it further.

Figure 2.12


2.4 The normal Mercator projection

Derivation of the Mercator projection

The generic function f () will be replaced by () for the normal Mercator projection.The condition that the projection is conformal, = , follows from Equation (2.17):

() =dd

= sec , (2.25)

and therefore() =

0

sec d , (2.26)

choosing a lower limit such that y(0) = R(0) = 0. The integrand may be rewritten using

cos = sin( +pi/2)= 2sin(/2+pi/4)cos(/2+pi/4)

= 2tan(/2+pi/4)cos2 (/2+pi/4)

so that (note [3] )

y = R() =R2

0

sec2 (/2+pi/4)tan(/2+pi/4)

d = R ln[

tan(2+pi4

)]. (2.27)

Therefore the Mercator projection in normal (equatorial) aspect is

x = Ry = R()

() = ln[

tan(2+pi4

)](2.28)

The following table gives the value of () at some selected latitudes. Note that sincetan(/2+pi/4) = cot(/2+pi/4) we must have () =().

() () ()

0 0.00 40 0.76 74.6 2

5 0.09 45 0.88 75 2.03

10 0.18 49.6 1 80 2.44

15 0.26 50 1.01 84.3 3

20 0.36 55 1.15 85 3.13

25 0.45 60 1.32 85.05 pi30 0.55 65 1.51 89 4.74

35 0.65 70 1.73 90

()

0 30 60 900

1

2

3

4

Figure 2.13


Mercator parameter and isometric latitude

The function () which occurs in the expression for the y-coordinate in the Mercator pro-jection is of importance in much that follows. We shall call it the Mercator parameter: thisis not standard usage although the term Mercator latitude was suggested by Lee (1946a).In advanced texts, such Snyder (1987), it is called the isometric latitude. But beware; otherauthors (see Adams, 1921) use the term for a different, but related, function. Note, too, thatthe symbol is not universal: Lee (1945) and Redfearn (1948) use , Maling (1992) use qand so on.

The term isometric latitude arises because the metric can be written in terms of and as ds2 =R2 cos2(d 2+d2). The coefficients in the metric are now both equal to R2 cos2so that equal increments of and correspond to the same linear displacements (at thelatitude concerned).

Having urged care with notation we must flag a small problem in notation. When wedefine the Mercator projection on the ellipsoid (NME in Chapter 6) we must define theMercator parameter in a slightly different way, but such that it reduces to (2.28) as theeccentricity e tends to zero. It would therefore have been natural to define the Mercator pa-rameter on the sphere as 0. We have not done this, assuming that the correct interpretationwill be obvious from the context.

Alternative forms of the Mercator parameter

The Mercator parameter can be cast into many forms which may be useful at times; herewe present five such. Consider the argument of the logarithm in (2.28):

tan(/2+pi/4) =1+ tan(/2)1 tan(/2) =

cos(/2)+ sin(/2)cos(/2) sin(/2)

=(cos(/2)+ sin(/2))2

cos2(/2) sin2(/2) =1+ sin

cos= sec + tan . (2.29)

Hence() = ln[sec + tan ]. (2.30)

Rearrange the penultimate term in (2.29):

1+ sincos

=

{(1+ sin)2

1 sin2

}1/2=

{1+ sin1 sin

}1/2.

Therefore

() =12

ln[

1+ sin1 sin

]. (2.31)

Exponentiate each side of (2.30) and then invert:

e = sec + tan ,e = sec tan ,


so that

2sinh = e e = 2tan ,and therefore

(a) sinh = tan , (b) sech = cos , (c) tanh = sin , (2.32)

from which we obtain further variants for ():

= sinh1 tan = sech1 cos = cosh1 sec = tanh1 sin . (2.33)

Inverse transformations and inverse scale factor

The inverse transformation for is trivial: = x/R (if 0 = 0). To find first set = y/Rand use the inverse of any of the expressions for given above. For example:

= 2tan1 e pi2= sin1 tanh = tan1 sinh = gd, =

yR

(2.34)

where we have introduced the gudermannian function gd defined below. (It may be recastinto many forms; see appendices at C.59, G.24 and web Weisstein, 2012):

gd(x) = x

0sech d = tan1 sinhx, gd1(x) =

x0

sec d = sinh1 tanx. (2.35)

The scale factor can also be considered as a function of the coordinates on the projection.Using Equation (2.32b) we have

k(x,y) = cosh = cosh(y/a). (2.36)

2.5 Rhumb lines and loxodromes

Rhumb lines were first discussed in the sixteenth century by Pedro Nunes as curves ofconstant azimuth which spiralled from pole to pole. The more academic word loxodrome(Greek loxos: oblique + dromos: running) appeared early in the seventeenth century. Atthat time both of these terms excluded simple parallel or meridian sailing but modern usageincludes these cases. This is unfortunate. We shall permit rhumb lines to include all possibledirections but we shall restrict the use of loxodrome to azimuths which are neither paral-lels nor meridians. This is consistent with the definition of the loxodrome in mathematicsas a spherical helix: see Weisstein (2012). The distinction is important for there are twotopologically distinct classes of rhumbs: (a) closed parallels; (b) open lines, loxodromes,running from pole to pole with meridians as degenerate cases. The importance of rhumblines follows from the conformality property: = implies that a rhumb line with constant is projected to a straight line on the Mercator projection.


-75

-60

-30

0

30

60

75

Figure 2.14: Rhumb line on the sphere and the Mercator projection

Figure 2.14 shows a loxodrome crossing the equator at 30W and maintaining a constantazimuth of 83: it spirals round the sphere covering a finite distance from pole to pole eventhough it makes an infinite number of turns about the axis. (These statements are provedbelow). On the projection, it is a repeated straight line of infinite total length as |y| .The intercepts with the Greenwich meridian are calculated later using Equation 2.43.

Figure 2.15: Infinitesimal elements of a rhumb on the sphere and the projection

Consider the rhumb distance, r12, on the sphere between P(1,1) and Q(2,2). Ifthe rhumb is a parallel the distance is the radius of the parallel circle times the change oflongitude. On a meridian the distance is simply the radius of the sphere times the change oflatitude. For an infinitesimal element of a loxodrome on the sphere, PQ in Figure 2.15a wehave cos = Rd/ds. Since is a constant this integrates trivially. In summary:

r12 = Rcos (21), parallel, (2.37)r12 = R(21), meridian, (2.38)r12 = Rsec (21), loxodrome. (2.39)

Therefore, to calculate the distance along a loxodrome, we need know only the constantazimuth and the change of latitude. This is an important result. Note (a) the meridian resultcan be deduced as the 0 limit of the loxodrome result (although the figure is inappro-priate); (b) the parallel result is not related to the loxodrome result by any limiting process.The latter is a reflection of the different topological nature of parallels and loxodromes.

The above equations show that the length of a loxodrome is finite. Setting 1 = pi/2and 2 = pi/2 in Equation 2.39 we obtain the total length from one pole to another aspiRsec . This reduces to piR on a meridian.


Equations of the loxodrome

To find the equation, on the sphere, of the loxodrome which starts at the point (1,1) at anazimuth note that on the projection it is the straight line

y y1 = (x x1) cot . (2.40)

where y1 = R(1) and x1 = R1. Using Equations 2.33 and 2.34 gives

() = (1)+( 1)cot , (2.41)

() = 1+ tan[

tanh1 sin tanh1 sin1], (2.42)

( ) = sin1tanh[tanh1 sin1+( 1)cot

]. (2.43)

As an example take 1 = 1 = 10 and = 2 = 2 = 40. Transforming to radianmeasure, Equation 2.41 gives

tan =pi

18050

[(40)(10)] = 0.929

and therefore = 42.9. Note that this is just the shortest rhumb line through the twopoints. If the rhumb makes one complete revolution before getting to the second point, thenreplacing = 50 by = 410 we find = 82.5 and so on.

Once has been determined further points on the rhumb are found from Equation 2.43.For example, for the loxodrome with = 83, 1 = 0 and 1 = 30 (Figure 2.14), wecalculate the intercepts on the Greenwich meridian ( = 0, 360, 720, . . .) as

3.7, 43.1, 67.3, 79.4, 85.0, 87.7, 88.9, 89.5, 89.8 . . ..

Equation 2.42 shows that becomes infinite at the poles so that the loxodrome mustencircle the pole an infinite number of times as it approaches, even though it is of finitelength. This is a geometrical version of Zenos paradox.

Mercator sailing

The above results solve the two basic problem of Mercator sailing, by which we meanloxodromic

mercator

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